Itô's Lemma: The Log Transform - Quant Trader Interview Question
Difficulty: Hard
Category: Stochastic Calculus
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Topics: Ito's Lemma, Stochastic Calculus, Geometric Brownian Motion, Log Transform
Problem Description
Suppose the price of a stock, $S_t$, follows a Geometric Brownian Motion (GBM) described by the following Stochastic Differential Equation (SDE):
$dS_t = \mu S_t dt + \sigma S_t dW_t$,
where $\mu$ is the constant drift, $\sigma$ is the constant volatility, and $W_t$ is a standard Brownian motion.
Using Itô's Lemma, determine the stochastic differential equation for $Y_t = \ln(S_t)$.
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